Generalized Cluster Algebra Homomorphisms Induced by Cluster Tube Embeddings

@misc{li2026generalized,
  title = {{Generalized Cluster Algebra Homomorphisms Induced by Cluster Tube Embeddings}},
  author = {Mingze Li},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.21822},
  eprint = {2610.21822},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.21822},
}

TY  - UNPB
TI  - Generalized Cluster Algebra Homomorphisms Induced by Cluster Tube Embeddings
AU  - Mingze Li
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.21822
UR  - https://arxiv.org/abs/2610.21822
AB  - We relate multiplication formulas for regular modules in coefficient-free acyclic affine cluster algebras with skew-symmetric exchange matrices to the imaginary exchange relations of Reading-Stella. Introducing a parameter z into the construction of Zhou-Zhu, we define a weighted cluster character on cluster tubes that satisfies generalized exchange relations of type C_n−1 and yields a bijection between indecomposable rigid objects and cluster variables. Using the multiplication formulas and difference property of Ding-Xiao-Xu, we show that non-homogeneous tube embeddings induce injective homomorphisms into ordinary affine cluster algebras under z↦ X_δ. Comparison with Reading-Stella's relations identifies X_δ with the corresponding theta function, giving a representation-theoretic interpretation of the generalized exchange parameter.
ER  -